ATP production and necrosis formation in a tumour spheroid model

Mathematical models of tumour spheroids, proposed since the early seventies, have been generally formulated in terms of a single diffusive nutrient which is critical for cell replication and cell viability. Only recently, attempts have been made to incorporate in the models the cell energy metabolism, by considering the interplay between glucose, oxygen and lactate (or pH). By assuming glucose and lactate as the only fuel substrates, we propose a simple model for the cell ATP production which takes into account the main reactions that occur in the glycolytic and the oxidative pathway. Under the assumption that cell death occurs when ATP production falls below a critical level, we have studied the free boundary problem for the concentration of glucose, lactate and oxygen inside the spheroid viable rim. We show that the existence of a necrotic core is guaranteed for a sufficiently large size of the spheroid.